where \(\varepsilon _{N_{i}}\)  is the \(CP\)  asymmetry parameter, \(\eta _{ii}\)  is the efficiency factor, while the numerical value in (REF ) depends on the number densities of RHNs over the entropy
density and on the sphaleron transitions, see [1]} for the details
yielding to this value.
The leading order terms given in eq. (REF ) induce higly suppressed \(CP\) 
asymmetry parameter \(\varepsilon _{N_{i}}\)  and thus a suppression of the
value of \(Y_{B}\) . Thus, we have to rely on NLO terms to obtain an
appropriate suppression of the \(CP\)  asymmetries. Therefore, we consider a case
in which the NLO contribution arises in the neutrino sector and
only corrects \(Y_{D}\)  in eq. (REF ). This is realized by
introducing a new flavon \(\omega \)  which transforms as \(1_{+-}\)  under \(D_{4}\)  with zero \(U(1)\)  charge
\(\delta W_{D}=\frac{y_{9}}{\Lambda }N_{3,2}^{c}F_{2,3}H_{5}\omega \) 
